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The formal failure and social success of logicWilliam BrookeUniversity of the Fraser Valley

Andrew Aberdein

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Zenker, F. (ed.). Argumentation: Cognition and Community. Proceedings of the 9th International Conference of the

Ontario Society for the Study of Argumentation (OSSA), May 18-21, 2011. Windsor, ON (CD ROM), pp. 1-7.

The formal failure and social success of logic

WILLIAM BROOKE

Department of Philosophy

University of the Fraser Valley

33844 King Road

Abbotsford, B.C.

Canada V2S 7M8

william.brooke@student.ufv.ca

ABSTRACT: Is formal logic a failure? It may be, if we accept the context-independent limits imposed by

Russell, Frege, and others. In response to difficulties arising from such limitations I present a Toulmin-

esque social recontextualization of formal logic. The results of my project provide a positive view of formal

logic as a success while simultaneously reaffirming the social and contextual concerns of argumentation

theorists, critical thinking scholars, and rhetoricians.

KEYWORDS: argumentation, dispute resolution, formal, Frege, Gödel, Leibniz, logic, peace, Russell

1. INTRODUCTION

Is formal logic a failure? If we judge the success of formal logic using criteria that are

imposed and evaluated from within that very field of inquiry, it seems that we are likely

to run into a number of difficulties. The aspirations of logicians have traditionally been

lofty, and their methods for achieving their many goals have typically been austere. Logi-

cians have developed systems that serve their attempts to transcend context, and to reach

towards the unchanging and eternal. The nature of the goals and methods of logicians

tend to obviate the consideration of formal logic as a social or historical entity. But de-

velopments in anthropology, sociology, and history—as well as work in new fields such

as argumentation theory and critical thinking—may demonstrate a value in placing in-

creased emphasis on those social and historical details that are normally considered ir-

relevant in accounts of the development of logic (e.g. Kneale & Kneale 1962). If we

abandon attempts to offer context-independent evaluations and begin to consider the his-

tory of formal logic in light of contextual concerns, it becomes possible to see—and more

importantly, to describe—logic as a success. To this end we will begin by investigating

some of the origins of formal logic in the 1600s, when logic held promise as an endeav-

our of social utility. We will then investigate the deterioration of social concerns within

the field of logic during the 1879-1931 period, address the all-too-common view of logic

as a ‘failure’, and conclude by seeing how the work of this period can be regarded as a

success when social factors are reintroduced.

2. LEIBNIZ AND HIS INFLUENCE

Toulmin (1989: 98-103) makes a compelling case for Leibniz’s development of a charac-

teristica universalis and calculus ratiocinator as a socially motivated activity. As evi-

denced by many of the some 15,000 or more letters written during his lifetime, Leibniz

WILLIAM BROOKE

2

was deeply troubled by the religious, economic, and political disputes of his day. While it

is possible to consider Leibniz and his contributions to logic in a relatively ahistorical

fashion, Toulmin argues that it is more sensible to see Leibniz as a “German intellectual

who accepted his responsibility to do whatever he could do to remedy the situation of

Europe in his time” (1989: 103). Leibniz makes clear reference to the resolution of real-

world disputes when proposing his universal calculus:

Whenever controversies arise, there will be no more need for arguing among two philoso-

phers than among two mathematicians. For it will suffice to take the pens into the hand and to

sit down by the abacus, saying to each other (and if they wish also to a friend called for help):

Let us calculate. (in Gabbay & Woods 2004: 1)

Properly formalized, details from philosophical, political, economic, and religious dis-

putes could be fed into a calculating system. Such a system would be supported by the

establishment of a global standard of communication, and would not only serve as a tool

of dispute resolution, but also as a universal solution to the barriers of language.

Leibniz’s proposed system may exhibit a “characteristic blend of genius and in-

sanity” (Coffa 1993: 14), but the influence of this idea in the development of logic has been

immense. In 1882, Ernst Schröeder became involved in a journal dispute with Gottlob Fre-

ge, who alleged that Schröeder had misunderstood the purpose of the Begriffsschrift. Frege

attempted to clarify this by responding: “I was trying, in fact, to create a ‘lingua character-

istica’ in the Leibnizian sense” (1968: 89). Soon thereafter, Peano and his followers took

up the habit of recalling Leibniz’s “project of creating a universal script ... in the opening

sentences of their general writings on logic” (Grattan-Guinness 2001: 237). Bertrand Rus-

sell—himself a notable Leibniz scholar—also wrote that “Leibniz foresaw the science

which Peano has perfected ... Leibniz’s dream has become fact” (1901; 1918: 79). The in-

fluences of Leibniz, Frege, and Peano eventually saw Russell, along with Alfred North

Whitehead, produce that great project of formal logic, the Principia Mathematica.

3. THE DEPARTURE OF THE SOCIAL

Leibniz had a definite influence on Frege’s logic. Frege’s Begriffsschrift was “influenced

by two of Leibniz’s ideas: a calculus ratiocinator (a formal calculus of reasoning) and a

lingua characteristica (a universal language)” (Moore, in Aspray & Kitcher 1988: 102).

According to Eike-Henner Kluge, the logical projects of Leibniz and Frege share “such

extensive and close correspondence on so many central issues that the hypothesis of a

fundamental influence on the part of Leibniz [is] very difficult to deny” (1980: 154). De-

spite similarities, Leibniz’s social motivations for his characteristica were not carried

forward by Frege. In the preface to his Begriffsschrift, Frege regards Leibniz’s “concep-

tion of a universal characteristic, a calculus philosophicus or ratiocinator, [as] too grandi-

ose for the attempt to realize it to go further than the bare preliminaries” (Heijenoort

1999: 5). In recognizing the scope and ambition of Leibniz’s project, Frege is also con-

scious that some two centuries have passed since Leibniz first envisioned his ‘dream’.

Yet Frege does not give up on Leibniz’s project altogether:

Even if this great aim cannot be achieved at the first attempt, one need not despair of a slow,

step by step approach. If the problem in its full generality appears insoluble, it has to be lim-

ited provisionally; it can then, perhaps, be dealt with by advancing gradually. Arithmetical,

THE FORMAL FAILURE AND SOCIAL SUCCESS OF LOGIC

3

geometrical, and chemical symbols can be regarded as realizations of the Leibnizian concep-

tion in particular fields. The Begriffsschrift offered here adds a new one to these—indeed, the

one located in the middle, adjoining all the others. (Heijenoort 1999: 6-7)

Frege can therefore be seen as continuing the work of Leibniz’s project, albeit in a lim-

ited manner. Frege saw clearly that diverse communities of academics such as geometers

and chemists had produced fragments of Leibnizian systems within their various disci-

plines. Taking care to describe his project as existing alongside, and not above or below

the piecemeal developments made in other disciplines, Frege plainly does not regard his

own project as a complete realization of Leibniz’s ‘dream’. He seems encouraged,

though, that the ‘central location’ of his own decidedly mathematical project might make

it the most promising path for timely advancement towards Leibniz’s goal.

The tremendous advancements made possible by Frege’s work during the fifty

years following the publication of the Begriffsschrift are testament to the effectiveness of

this limitation of scope. In particular, Frege’s project paved the way for that two-thousand

page behemoth rumoured to have arrived at Cambridge University Press in a wheelbar-

row, the Principia Mathematica. While the Principia made great inroads in the logicist

project, the social concerns of Leibniz were nowhere to be seen. Like Leibniz, Russell

was deeply concerned with peace. This concern was certainly present following the Boer

war, and persisted throughout the remainder of Russell’s life. But for Russell, logic had

become l’art pour l’art. Above all else, the Russell of this period was committed to the

strictly academic improvement of intellectual discipline and the pursuit of clarity, objec-

tivity, and truth. It seems possible that Russell might have conceived of such goods as

intrinsic benefits that might stem from study of the Principia. However, Russell is not

recalled for any insistence that some socially pragmatic value might be attached to his

efforts in logic, and, most certainly, not in the sense suggested by Leibniz.

4. GÖDEL: MATHEMATICS AND MISAPPROPRIATION

The path of advancement that was opened by Frege’s focus on logicism and continued by

the work of Russell slowed in 1931, when Kurt Gödel presented his incompleteness theo-

rem. The projects of Frege and Russell sought to place mathematics on firm logical foun-

dations, but Gödel’s incompleteness theorem showed that such systems were unable to

capture all of the “truths of basic arithmetic” (Smith 2007: 122). Among modern academ-

ics, the popular picture that is painted of the 1879-1931 period in formal logic is often

unfavourable. The rationale for these views seems to arise from a kind of narrative in

which Frege plays the role of ‘alpha’, while Gödel figures as ‘omega’. For example, Ian

Hacking characterizes logicism as an “arithmetical revolution [that] failed to take place”

(2000: 45) and describes the Principia as a project that “did not pan out, for very famous

reasons, connected with the name of Kurt Gödel” (2000: 45). Similar views are not re-

stricted to philosophers:

In the Principia [Whitehead and Russell] took up, among other tasks, the project of establish-

ing two theses about logic and mathematics: first, that the disciplines can be complete and,

second, that they can be consistent. ... The Principia, a logocentric enterprise par excellence,

seeks to establish the self-sufficiency and integrity of such concepts as identity, contradiction,

proof, and system. Whitehead and Russell failed in this effort ... it remained to Gödel to show

them why. (Thomas 1995: 250)

WILLIAM BROOKE

4

Gödel’s result showed that the logical systems of Frege and Russell were incapable of

providing axiomatic foundations from which the entirety of mathematics could be de-

rived. But quotations such as the preceding make too much of Gödel’s result, and, in so

doing, run the risk of excessively devaluing the importance of formal logic. Such views

are not always discouraged by the language use of mathematicians and logicians, who

have described the impact of Gödel’s result with dramatic adjectives. Michael Friedman

characterizes Gödel’s theorem as a “fatal blow” (in Aspray & Kitcher 1988: 93) for the

logical projects of the 1879-1931 period. Similarly, Lindstrom and Palmgren suggest that

Gödel’s result “shattered” (2010: 19) such projects, while Sinaceur uses the term “de-

stroyed” (2010: 376). The efforts of logicians through the 1879-1931 period have even

been branded as a quest for the “Holy Grail of logic” (Girle 2003: 192; Hunter 1996: 93);

in such a narrative, the incompleteness theorem produced by Gödel stands as the central

obstacle preventing the acquisition of the ‘grail’.

To these characterizations we must add the realization that Gödel’s result has

become the subject of frequent and problematic misappropriations in more popular ven-

ues of discourse. Generally speaking, these troubling examples involve the misapplica-

tion of Gödel’s result to subjects beyond the realm of mathematics. Examples of these

misuses can be found in philosophy: as early as 1958, William Barrett wrote, “if human

reason can never reach complete systematization in mathematics, it is not likely to reach

it anywhere else” (p. 39). In The Postmodern Condition, Jean-François Lyotard sees Gö-

del’s result as necessitating a “reformulation of the question of the legitimation of

knowledge” (1984: 43). Elsewhere, Gödel’s result has been compared to Zen Buddhist

kōans (Hofstadter 1980: 246-272), and used to offer an explanation of the nature of hu-

man consciousness (Penrose 1996: 64-126). Each of these examples takes Gödel’s result

far beyond the realm of mathematics. In a world where Gödel’s result is inexpertly in-

voked in discussions ranging from theology to politics, one academic has even seen fit to

produce a guide to the use and abuse of Gödel’s theorems. Another introductory philoso-

phy text suggests that people who cite Gödel’s result should be “assumed guilty until

proven innocent” (Baggini & Fosl 2010: 252).

5. WHEN ‘FAILURE’ IS SUCCESS

From a mathematical standpoint, Gödel’s theorem is a particularly valuable result. In and

of itself, it offers no reason to regard formal logic—or the Principia, as the paradigmatic

representative of efforts in formal logic—as failures. Indeed, on formal grounds, it seems

more correct to say that the projects of formal logic are enriched, and not destroyed or

shattered, by Gödel’s result. B.J. Sokol writes: “in mathematics, a negative conclusion

has always been as richly interesting as a positive one” (1995: 1054). This positivity has

been embraced by some writers, who offer logical or mathematical descriptions of the

impact of Gödel’s result that do not engage in overly dramatic exaggerations:

There’s an old hope (which goes back to Leibniz) that can be put in modern terms like this:

we might one day be able to mechanize mathematical reasoning to the point that a suitably

primed computer could solve all mathematical problems in a domain by deciding theo-

remhood in an appropriate formal theory. What [Gödel has] shown is that this is a false hope.

(Smith 2007: 45)

THE FORMAL FAILURE AND SOCIAL SUCCESS OF LOGIC

5

While the preceding description does not overstep the scope of Gödel’s result, it serves as

an example of the inaccessibility of logical or mathematical discourse. Regrettably, accu-

rate descriptions of Gödel’s result that are presented by mathematicians and logicians

seem to fall largely upon deaf ears. When such accounts are compared with the easily

accessible and hyperbolic alternatives offered by other writers, it is perhaps unsurprising,

although unfortunate, that Gödel’s result is so widely misunderstood.

Understanding formal logic as a socially situated and pragmatically motivated

activity allows us to generate a semantic alternative to the normal mathematical and logi-

cal explanations for Gödel’s result and its implications for the projects of formal logic.

This semantic alternative makes use of the vocabulary of argumentation theorists and

critical thinking scholars, and centres on the idea of viewing formal logic as a socio-

political attempt at dispute resolution and peacemaking. If we invoke Toulmin’s under-

standing of Leibniz as a socially motivated individual who sought a system by which

peace and understanding could more easily be established, then Gödel’s result should not

be regarded as the end of the road for Leibniz’s vision, nor as dealing a ‘death blow’ to

the ‘failed’ projects of formal logic. Rather, what Gödel’s result has shown is that Leib-

niz’s dream of a system by which international peace could be established and maintained

will not be fulfilled in a singularly mathematical manner; it does not preclude the possi-

bility that Leibniz’s goal might be achieved in some other fashion:

Gödel’s theorem, far from undermining the project of Principia, provided its culminating

achievement and glory, by showing that mechanical decidability, the mere grinding out, of

significant mathematical proofs is impossible. Unaided computers, for instance, will never be

able to prove (or disprove) weighty mathematical theorems, no more than ruler and compass

will be able to trisect angles. (Sokol 1995: 1054)

If anything, Gödel’s theorem should leave us uncertain as to the pragmatic status of

mathematical problem solving. Along with the efforts of Frege and Russell, Gödel’s theo-

rem shows us that we would be well advised to maintain diligence and rigor in any analy-

sis of argumentation and in the resolution of disputes, even—and perhaps especially—if

no mathematical ‘autopilot’ exists for the resolution of disagreements. This very fact of

knowing that mathematical reasoning may be insufficient for the successful resolution of

disputes is incredibly significant, and recommends the investigation of other, more social-

ly and contextually robust, forms of analysis and reasoning. At core, an explanation using

these semantics does not differ significantly from the mathematical and scientific idea

that negative and positive results can be of equal value. It may be the case, however, that

these semantics are able to communicate the efforts of formal logicians and the result

found by Gödel with greater explanatory power.

Let us remain sceptical, though, and proceed with care, so as to ensure that the

use of these semantics does not easily result in the abuse of Gödel’s theorem. With these

semantics, let us hypothesize three highly generalized possible scenarios in which we

might consider the immediate comprehensibility of likely and potential implications for

our socio-political view of formal logic. First, suppose that a system of peacemaking and

dispute-resolution such as the one envisioned by Leibniz has in fact become a reality. In

this scenario, every individual might have a handheld device with an ‘app’ for the near-

instantaneous resolution of all matters of disputes, philosophical or otherwise. In a se-

cond, far more negative situation, we hypothesize that a fundamental meta-theorem im-

pacting all areas of human activity and thought has proven beyond any doubt that all sys-

WILLIAM BROOKE

6

tems of dispute-resolution are doomed to failure. In a third, neutral situation, we can im-

agine that a theorem specific to one field of human inquiry has strongly suggested that a

particular mode of human activity might not be able to provide a concrete, failsafe system

for the resolution of disputes, but that other modes of human activity might still be used

to achieve peace and prosperity. By analogy, it seems that the second, negative scenario

has thus far been the most common interpretation of Gödel’s result. However, this nega-

tive scenario is not supported by our intuitions: personal experience readily suggests that

most people are clearly able to use systems—such as computers—to solve problems. The

positive situation, too, finds little in the way of empirical support: it is uncontroversial

that people do not have portable handheld applications that are capable of resolving all

problems; indeed, our lives are riddled with ongoing disputes. The third scenario, then,

seems to not only best describe Gödel’s result, but also appears to be the only intuitively

appealing option in terms of real-world practice.

6. CONCLUSION

Abandoning an ahistorical view of formal logic encourages the consideration of socio-

political motivations in the study of notable logicians and their works. Such studies per-

mit the usage of a new set of socially germane semantics, which may allow the efforts of

formal logicians to be more accurately communicated to non-logicians. The value of a

socially, historically, and contextually situated view of formal logic may be contested.

However, it seems naïve to ignore the many social factors that motivated various devel-

opments in the works and goals of formal logicians. As Toulmin suggests, Leibniz might

well have devised his characteristica in order to address deeply troubling social concerns.

It seems clear, too, that such social concerns fall out of sight in the work of Frege and

Russell, whose aspirations for logic shifted towards those abstract and decontextualized

goals with which we are now so familiar. But Russell and Frege need not have offered

explicit endorsements regarding the pragmatic values of their works in order for us to

acknowledge their social and real-world applications. Even intensely abstract and seem-

ingly asocial efforts are of social consequence: consider the physicists of the Manhattan

Project, whose protracted unawareness of the profound and inescapable implications of

their work did not lessen the real-world impact of atomic weapons.

Formal logicians may experience some measure of discomfort with the present-

ed view of formal logic as an effort of socio-political dispute resolution. They may prefer

to preserve a pure view of their field of study, and maintain that their goals for formal

logic transcend context. If, in time, it becomes apparent that viewing formal logic as a

socially situated and pragmatic activity will allow other academics to have a more accu-

rate—and more positive—view of formal logic, then perhaps such semantics will curry

favour with formal logicians. For philosophers concerned with argumentation theory and

the development of critical thinking skills—i.e., those subfields broadly regarded as ‘in-

formal logic’—such an account recommends a more positive view of the history of for-

mal logic, as well as the herculean efforts of formal logicians. At core, the idea of seeing

formal logic as an effort in dispute-resolution is not at odds with the values of ‘informal’

practitioners. In conclusion, we may therefore be mindful that formal and informal practi-

tioners share many of the same goals, and that, if these groups disagree, it is perhaps

mainly in their opinions regarding how those goals might be reached.

THE FORMAL FAILURE AND SOCIAL SUCCESS OF LOGIC

7

ACKNOWLEDGEMENTS: I would like to thank Andrew Aberdein, Anastasia Ander-

son, Glen Baier, James Bradley, Jack Brown, Dustin Ellis, Scott Fast, Nicholas Griffin,

Paul Herman, Dale Jacquette, Moira Kloster, Jeffrey Morgan, Philip Rose, Nadeane

Trowse, and John Woods for their helpful comments.

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Frege, G. (1968). On the purpose of the Begriffsschrift. Australasian Journal of Philosophy, 46(2), 89.

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Russell, B. (2009). Mysticism and Logic: And Other Essays. Cornell University Library.

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Thomas, D. W. (1995). Godel's theorem and postmodern theory. PMLA: Publications of the Modern

Language Association of America 110(2), 248.

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Ontario Society for the Study of Argumentation (OSSA), May 18-21, 2011. Windsor, ON (CD ROM), pp. 1-5.

Commentary on “THE FORMAL FAILURE AND

SOCIAL SUCCESS OF LOGIC” by William Brooke

ANDREW ABERDEIN

Humanities and Communication,

Florida Institute of Technology,

150 West University Blvd, Melbourne, Florida 32901-6975

U.S.A.

aberdein@fit.edu

1. INTRODUCTION

I am grateful to the organizers for giving me the opportunity to reply to William Brooke’s

scholarly and thought-provoking paper. Brooke traces the influence of Leibniz’s concept of

a universal language and mechanical system of rational dispute resolution, the characteris-

tica universalis, to Frege and Russell, and ultimately to Gödel. Gödel’s incompleteness re-

sults are often, but erroneously, seen as exploding Leibniz’s project. Brooke argues that

they only have this effect if the characteristica universalis is understood in formal, mathe-

matical terms. He contends that an informal characteristica universalis—a source of dis-

pute resolution through non-mathematical means—is still open to us; and as informal logi-

cians we should be well placed to take up where Leibniz left off. This narrative has conse-

quences in several fields. I shall concentrate on three: politics, mathematics, and logic.

2. POLITICS

Gottfried Wilhelm Leibniz was born in 1646, two years before the end of the Thirty Years

War, by far the most destructive European conflict prior to the twentieth century. Through-

out his life he was an indefatigable diplomatist. The pursuit of peace is, as Brooke ob-

serves, an unsurprising priority for his philosophical work. The means by which he thought

it could be achieved, a system of reasoning in which all controversies might be resolved by

mechanical means, is surprising. Nonetheless, Brooke follows Toulmin in arguing persua-

sively that these social concerns were the key motivation for Leibniz’s logical work.

Bertrand Russell’s work in logic far surpassed that of Leibniz in technical so-

phistication. And Russell is also remembered as an outspoken advocate of peace, from his

pacifism in World War I, to his advocacy of nuclear disarmament. However, as Brooke

notes, ‘Russell is not recalled for any insistence that some socially pragmatic value might

be attached to his efforts in logic, and, most certainly, not in the sense suggested by Leib-

niz’ (Brooke 2011: 3). Indeed, Russell is scathing about the prospects for the characteris-

tica universalis, ascribing to Leibniz a fundamental confusion about the location of philo-

sophical problems:

For the business of philosophy is just the discovery of those simple notions, and those primi-

tive axioms, upon which any calculus or science must be based. … And thus Leibniz sup-

posed that the great requisite was a convenient method of deduction. Whereas, in fact, the

problems of philosophy should be anterior to deduction. (Russell 1900: 201)

ANDREW ABERDEIN

2

Thus, for Russell, the hard problem of philosophy is finding the right principles; reasoning

from these principles is easy by comparison. Russell’s position is echoed by John Woods in

his identification of Philosophy’s Most Difficult Problem: ‘that of adjudicating in a princi-

pled way the conflict between supposing that [a valid argument] is a sound demonstration

of a counterintuitive truth, as opposed to seeing it as a counterexample of its premisses’

(Woods 2003: 14). This problem remains, and remains difficult, no matter what progress

we make in realizing Leibniz’s characteristica universalis, whether formally or informally.

Nonetheless, Russell does acknowledge the practical utility of informal reason-

ing. For example, in a letter published in the February 16th, 1941 edition of the New York

Times, Russell offered a closely reasoned explanation for his support for the British war

effort, and argued that the United States should enter World War II. In the course of his

argument, he reflects upon the style of reasoning it employs:

The whole argument, either way, is necessarily based on hypotheses and probabilities, as to

which no certainty is possible; but as immediate decisions are forced upon us, we have to act

upon such data as can be obtained, knowing that even the most careful consideration may

lead us astray. (Russell 2002: 182)

Hence, although Russell may have seen no application for his own formal logic in arguments

over war and peace, he does acknowledge a role for what we now call informal logic.

The famously unworldly Kurt Gödel had little interest in politics. However, he

was more sanguine about the prospects for the characteristica universalis than Russell.

Indeed, he seems to have believed that Leibniz had actually perfected the scheme:

But there is no need to give up hope. Leibniz did not in his writings about the Characteristica

universalis speak of a utopian project; if we are to believe his words he had developed this

calculus of reasoning to a large extent, but was waiting with its publication till the seed could

fall on fertile ground. (Gödel 1944: 152 f.)

Taken literally, as Martin Davis notes, this seems quite mad: Leibniz’s logic is not re-

motely sophisticated enough to accomplish this goal (Davis 2000: 134). However, Gö-

del’s interest in the characteristica universalis does seem to have been restricted to its

mathematical workings, rather than its political ambitions. It is the relationship between

Gödel’s own mathematical work, specifically his incompleteness results, and Leibniz’s

project that most concerns Brooke.

3. MATHEMATICS

Gödel’s incompleteness results are of profound importance, but precisely stating that im-

portance can be a challenge. Brooke rightly deprecates the ‘application of a misunder-

stood rendition of Gödel’s result to subjects beyond the realm of mathematics’, and ob-

serves that there is a great temptation to ‘make too much of Gödel’s result, and, in so do-

ing, run the risk of excessively devaluing the importance of formal logic’ (Brooke 2011:

4). As Jon Barwise put it, ‘Karl Marx is supposed to have said, “Every time the train of

history goes around a corner, the thinkers fall off.” Gödel’s Theorem was a very sharp

corner on the logic line’ (Barwise 1981: 99). Brooke itemizes some of the logic line’s

more accident prone passengers. But this sort of critique is infectious. Brooke tells us that

‘what Gödel’s result has shown is that Leibniz’s dream of a system by which internation-

al peace could be established and maintained will not be fulfilled in a solely mathematical

COMMENTARY

3

manner’ (Brooke 2011: 5). While international peace may be unattainable for many rea-

sons, Gödel’s result does not strictly place the characteristica universalis beyond the

scope of mathematics, but rather of any single formal system within mathematics. As

Martin Davis summarizes,

Gödel’s incompleteness theorem shows that ... [f]or any specific given formalism there are

mathematical questions that will transcend it. On the other hand, in principle, each such ques-

tion leads to a more powerful system which enables the resolution of that question. One envi-

sions hierarchies of ever more powerful systems each making it possible to decide questions

left undecidable by weaker systems. (Davis 2000: 124)

On this picture, mathematics is unbounded, but at the expense of outgrowing every for-

mal system. As Raymond Smullyan concludes,

In the prophetic words of the logician Post, this means that mathematics is, and must remain,

essentially creative. Or, as commented by the mathematician Rosenbloom, it means that man

can never eliminate the necessity of using his own intelligence, regardless of how cleverly he

tries. (Smullyan 2001: 88)

In other words, mathematics can never fully transcend its informal aspects. This suggests

that the study of informal reasoning should play an indispensable role in the analysis of

mathematical practice (as I have argued elsewhere, for other reasons: see, for example,

Aberdein 2009).

4. LOGIC

Brooke concludes with the salutary reminder ‘that formal and informal practitioners share

many of the same goals, and that, if we disagree, it is mainly in our opinions regarding

how those goals might be reached’ (Brooke 2011: 5). I want to conclude with a few

words about the nature of that disagreement. Elsewhere I have characterized the broader

context required for a system of logic to be advocated as applicable to natural argumenta-

tion as a logical theory, a quadruple comprising

system, parsing theory, inferential goal, background theories

(Aberdein and Read 2009: 618; cf. Thagard 1982: 37). Each of these components needs a

little explication. The system comprises the syntax, semantics, and metatheory of the logic,

or whatever other such details its presentation requires. The parsing theory consists of a

method for translating between the system and natural, unsystematized usage. The inferen-

tial goal determines what the use of this logic is intended to achieve and what its valid in-

ferences are expected to preserve. The background theories are broader philosophical or psy-

chological theories which inform the choices made in putting together the rest of the theory.

This account gives rise to a hierarchy of moves by which a logical theory may be

revised in response to a problem:

ANDREW ABERDEIN

4

I Indifference;

II Non-revisionary responses:

(a) Delimitation of the subject matter of logic;

(b) Novel paraphrase;

(c) Semantic innovation;

III Conservatively revisionary responses;

IV Non-conservatively revisionary responses:

(a) Restriction of the logic;

(b) Wholesale revision;

V Change of subject matter

(cf. Aberdein and Read 2009: 635).

Brooke draws attention to a problem which became increasingly acute for formal logic in

the mid-twentieth century: its inability to contribute significantly to such applied prob-

lems as socio-political dispute resolution. One response was the development of argu-

mentation theory by such pioneers as Toulmin and Perelman. In terms of the above hier-

archy, this exemplifies the last and most radical of these moves: a change of subject mat-

ter. This occurs when a change of inferential goal is ‘precipitated by a non-conservative

revision of the background theories. Typically this will alter the motivation of the whole

logical enterprise, move the problem into a different area, and change the subject matter

of logic’ (Aberdein and Read 2009: 640). This coincides with, for example, the stance of

Ralph Johnson and Tony Blair, who ‘distinguish informal logic from formal logic, not

only by methodology but also by its focal point ... the cogency of the support that reasons

provide for the conclusions they are supposed to back up’ (Johnson and Blair 1997: 161).

In my terminology this focal point is the inferential goal. A different response emerged

somewhat later with the development of systems of non-classical formal logic, such as

non-monotonic logic, which were better adapted to the vicissitudes of natural argumenta-

tion. These systems exemplify the previous level of the hierarchy: wholesale, non-

conservative revision of the system, accompanied by appropriate adaptations to the rest of

the logical theory. For several decades these two responses to the same problem proceed-

ed independently, and indeed, in apparent ignorance of each other. But more recently, we

have begun to see a rapprochement between the two. It is perhaps in such hybridization

that Leibniz’s goal stands the best hope of some form of fulfilment.

COMMENTARY

5

REFERENCES

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